Introduction to Real Analysis and Fourier Analysis Mijia Lai updated on March 8,2020
Introduction to Real Analysis and Fourier Analysis Mijia Lai updated on March 8, 2020
Contents 55 1.2 Cardinality 1.3 Topology of the Euclidean space 5c and Digory the 901 1.5.1 Hausdorff distance and Gromoy-Hausdorff distance 1.5.2 Invariant of domain............... 2 Lebesgue measure 2.1 Exterior measure 。。 asurable ets 92 2.5 Sets of positive measure 3.1 urable 41 3.2 Simple functions 3.3 Littlewood'sThree principles,.,,,,,,, 30 4 Lebesgue's integration theory 4.】Integration nterchanging limits with integrals 4.4 Fubini's Theore 5 Differentiatic 5.2 Fundamental theorem of Calculus I 5.2.1 A detour:Bounded variation funetions......................50 6 Function spaces 6.1 6.11 6.12 A detour:Conveity ad Jense'inquaty 6.1.3 Completeness:Banach space... 62 3
Contents 1 Preliminary 5 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Topology of the Euclidean space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 Metric space and Baire Category theorem . . . . . . . . . . . . . . . . . . . . . . . . 10 1.5 Continuous functions and Distance in metric space . . . . . . . . . . . . . . . . . . . 11 1.5.1 Hausdorff distance and Gromov-Hausdorff distance . . . . . . . . . . . . . . . 13 1.5.2 Invariant of domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2 Lebesgue measure 17 2.1 Exterior measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3 Borel sets and Measurable sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4 Linear transformation of measurable sets . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.5 Sets of positive measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3 Measurable functions 27 3.1 Measurable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Simple functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.3 Littlewood’s Three principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4 Lebesgue’s integration theory 33 4.1 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.2 Interchanging limits with integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.3 Lebesgue v.s. Riemann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.4 Fubini’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5 Differentiation 45 5.1 Monotone functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.2 Fundamental theorem of Calculus I . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.2.1 A detour: Bounded variation functions . . . . . . . . . . . . . . . . . . . . . . 50 5.3 Fundamental theorem of Calculus II . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.4 Lebesgue Differentiation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 6 Function spaces 59 6.1 L P spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 6.1.1 Normed vector space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6.1.2 A detour: Convexity and Jensen’s inequality . . . . . . . . . . . . . . . . . . 61 6.1.3 Completeness: Banach space . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 6.1.4 Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3
4 CONTENTS 6.2H 62.10pac9 6.2.2 6 6.2.3 Linear functional.Duality............ 5 7 Fourier Series 7.1 ntroduction· 7202 twise convergence 7.2.2 Abel summation 072 7.3 L2 convergence 7.4 Applications 72 8 Fourier Transforms 8.1.1 Fourier transform on S(R) 777 8.1.2 Inversion formula 82 rm on 8.3.1 Heat equation on R 9 Selected topics 9. 11 finite 912 Euler product formula 9.2 Falconer conjecture..... 88858888 92.4 Fourier transform to measure 93 of large numbers and Centra limit theorem 。 9.3.3 Central limit theorem
4 CONTENTS 6.2 Hilbert space: L 2 spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 6.2.1 Inner product and Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . 63 6.2.2 Orthogonality, Orthonormal basis, Fourier series . . . . . . . . . . . . . . . . 64 6.2.3 Linear functional, Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 7 Fourier Series 69 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 7.2 Pointwise convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 7.2.1 Ces`aro summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 7.2.2 Abel summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 7.3 L 2 convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 7.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 7.4.1 Isoperimetric inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 7.4.2 Weyl’s equidistribution theorem . . . . . . . . . . . . . . . . . . . . . . . . . 74 8 Fourier Transforms 77 8.1 Fourier transform on R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 8.1.1 Fourier transform on S(R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 8.1.2 Inversion formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 8.1.3 The Plancherel formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 8.2 Fourier transform on R n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 8.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 8.3.1 Heat equation on R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 8.3.2 Harmonic functions on upper half plane . . . . . . . . . . . . . . . . . . . . . 82 8.3.3 Wave equation in R n × R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 9 Selected topics 83 9.1 Dirichlet Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 9.1.1 Fourier analysis on finite group . . . . . . . . . . . . . . . . . . . . . . . . . . 84 9.1.2 Euler product formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 9.2 Falconer conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 9.2.1 Hausdorff measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 9.2.2 Falconer conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 9.2.3 Abstract Borel measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 9.2.4 Fourier transform to measure . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 9.3 Law of large numbers and Central limit theorem . . . . . . . . . . . . . . . . . . . . 90 9.3.1 A crash course in probability . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 9.3.2 Law of large numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 9.3.3 Central limit theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Chapter 1 Preliminary 行路难！行路难！多歧路，今安在？ 长风硫浪会有时直辈否膏露潜 1.1 Introduction This lecture note is prepared for the course Introduction to Real analysis and Fourier analysis.It can into two parts.T hrst part the tegatioi number of discontimous pointsis countable.Therefore the integr almost continuous functions.Even though the great triumph was achieved by the Riemannian stha Indeed continuous functions are convergent to f then 1.f may not be Riemannian integrable; 2.even f is Riemannian integrable, 。in(h=厂feld may not hold. We give a counter-example for item 1 in the above.We can enumerate all rational numbers in [0,1]as (a,q2.........define 1a)-{bm0…9 It follows that f converges to the Dirichlet function D(),which is not Riemannian integrable. 5
Chapter 1 Preliminary 1¥Jú1¥Júı‹¥ß8S3º ºªL¨kûßÜ! ~LÙ°" ))ox51¥J6 1.1 Introduction This lecture note is prepared for the course Introduction to Real analysis and Fourier analysis. It can be roughly divided into two parts. The main subject in the first part is the Lebesgue’s integration theory. We have learned in Calculus that a function is Riemannian integrable if and only if the number of discontinuous points is countable. Therefore the Riemannian integral mainly works with almost continuous functions. Even though the great triumph was achieved by the Riemannian integral, it still has a major defect: not working well with limit. Indeed, continuous functions are not closed under taking limit, i.e., the limit of sequence of continuous functions is not necessarily continuous. Moreover, let fn be a sequence of Riemannian integrable functions on [0, 1], which is convergent to f then 1. f may not be Riemannian integrable; 2. even f is Riemannian integrable, limn→∞ Z 1 0 fn(x)dx = Z 1 0 f(x)dx may not hold. We give a counter-example for item 1 in the above. We can enumerate all rational numbers in [0, 1] as {q1, q2, · · · , ...}, define fn(x) = 1, x = q1, q2, · · · , qn; 0, else. It follows that fn converges to the Dirichlet function D(x), which is not Riemannian integrable. 5
6 CHapTER L PRELIMINARY in vals (cubes). underlining Euclidean geometry,on the other hand,put strong restrictions onto the local behavior of emanian integral represents theare e curve,th strips.Le horizontal strip may spread everywhere,however,it turns out to be a sweet surprise.As the local behavior of th in consideratio e care This viewpoint dramatically enlarges the range of integrable functions.The corresponding inte gral theory now boils down to the definition of the measure,and the rest follows alm nost naturally Another great advantage of reat convenience in subiect such as probability theory. -wise,in this course we shall provide the following generalization: length,.area,volume,…→measure continuous functions =measurable functions Riemannian integral=Lebesgue integral n the following,we sketch some important historical moments of the development for the real
6 CHAPTER 1. PRELIMINARY The basic idea of Riemannian integral is to divide the domain of definition into small intervals (cubes for higher dimensions). These neighboring intervals (cubes), on the one hand, rely on the underlining Euclidean geometry, on the other hand, put strong restrictions onto the local behavior of integrable functions. (cannot oscillate too much, thus leading to the continuity to some extent) The geometric meaning of the Riemannian integral represents the area under the curve, thus Riemann’s way of integration, roughly speaking, is to approximate the area by dividing the region into vertical strips. Lebesgue’s viewpoint is to view the region by horizontal strips. At a first glance, each horizontal strip may spread everywhere, however, it turns out to be a sweet surprise. As the local behavior of the function in consideration is not so critical, and what really matters now is the set of the form {f ≥ c}, which motivates the careful definition of its measure (strictly speaking, in this book by measure we mean Lebesgue measure). This viewpoint dramatically enlarges the range of integrable functions. The corresponding integral theory now boils down to the definition of the measure, and the rest follows almost naturally. Another great advantage of Lebesgue’s integral theory is that it is not restricted only to the integration on Euclidean space. It can equally be transplanted to any abstract measure space, yielding great convenience in subject such as probability theory. We shall see the above counter-example holds true in the sense of Lebesgue’ integration. Namely, the Dirichlet function is Lebesgue integrable and our hope that limn→∞ R [0,1] fn(x)dx = R [0,1] D(x)dx becomes true. Vocabulary-wise, in this course we shall provide the following generalization: length, area, volume, ... =⇒ measure continuous functions =⇒ measurable functions Riemannian integral =⇒ Lebesgue integral In the following, we sketch some important historical moments of the development for the real analysis